1 Introduction
Let
$\Gamma (1)=SL_2({\mathbb {Z}})$
act on the upper-half plane
$\mathbb {H} = \{z\,|\,\mathrm {Im}z>0\}$
by the Möbius transformation
Two points in
$\mathbb {H}$
are called equivalent under
$\Gamma <\Gamma (1)$
if they are in the same orbit of
$\Gamma $
. We have the following definition from [Reference ShimuraShi71, p. 15].
Definition 1.1 For a discrete subgroup
$\Gamma <\Gamma (1)$
, we call a set
$F\subset \mathbb {H}$
a fundamental domain for
$\Gamma $
if
-
(1) F is open and connected;
-
(2) no two points of F are equivalent under $\Gamma $
; -
(3) every point of $\mathbb {H}$
is equivalent to a point in
$\overline F$
, the closure of F.
It is well known that a fundamental domain for
$\Gamma (1)$
in the above sense is
Since
$-I$
is in the center of
$\Gamma (1)$
, we see that
$(\pm I) \Gamma $
is also a subgroup of
$\Gamma (1)$
. When
$-I\in \Gamma $
,
$(\pm I) \Gamma =\Gamma $
. Note that the Möbius transformation for
$-I$
is the identity. If we have a choice of right coset representatives
where
$n=[\Gamma (1):(\pm I)\Gamma ]$
, we will show in Lemma 2.1 that
is a fundamental domain for
$\Gamma $
provided that it is connected, where the closure is, as always in this article, taken in
$\mathbb {H}$
. We will then call such a list of right coset representatives
$ \{\gamma _1,\dots ,\gamma _n\} $
fundamental.
Consider the congruence subgroups [Reference Diamond and ShurmanDS05, p. 13]
The main purpose of this article is to provide explicit and canonical lists of fundamental right coset representatives such that the corresponding domains are connected.
There are several classical and computational approaches to fundamental domains for congruence subgroups. Chuman gave an explicit Schreier system of representatives for
$\Gamma _0(N)\backslash \Gamma (1)$
, with applications to generators and relations of
$\Gamma _0(N)$
[Reference ChumanChu73]. More generally, Kulkarni’s Farey symbol method gives an arithmetic-geometric framework for constructing special polygons for finite-index subgroups of the modular group [Reference KulkarniKul91]. There are also computer programs for finding and drawing connected fundamental domains, including Verrill’s program [Reference VerrillVer01]. The emphasis here is different. We give a closed-form family of representatives whose connectedness is built into the construction. The resulting domains provide a concrete entry point to the geometry of modular curves and complement the cusp-class and Farey symbol approaches of Chuman and Kulkarni.
It is also useful to view our domains at a slightly coarser level. Each standard triangle has three distinguished sides, which in the figures may be called the left side, the right side, and the base side. If
$\gamma D$
is one of our standard triangles, then the triangle across its base side is
$\gamma S D$
. Thus, whenever both
$\gamma $
and
$\gamma S$
occur in our representative list, the common base side is an internal edge and the two triangles form a two-triangle cell, or kite. In this sense, the domains constructed below may be regarded first as triangulated domains and then, after internal base-side gluings, as partially decomposed into kites. This gives a visual bridge between our explicit standard-tile domains and the coarser polygonal regions appearing in Farey symbol constructions. We do not claim that this provides a new construction of Kulkarni special polygons, as the Farey symbols have become well established and easy to use [Reference Kurth and LongKL08].
In this article, we mainly work in the ring
${\mathbb {Z}}/N{\mathbb {Z}}$
. We write the elements as if they are just integers as
$a, b\in {\mathbb {Z}}/N{\mathbb {Z}}$
. Note that the greatest common divisors
$\gcd (a, N)$
and
$\gcd (a, b, N)$
are well defined as positive integers between 1 and N.
To make our final pictures more centered and hence more visualizable, we define our symmetric choice of residue classes mod N. The usual choice of classes from
$0$
to
$N-1$
mathematically would work the same (with suitable small modifications), but the pictures would not be as nice when drawn out.
Notation 1.5 For
$N>1$
, let
$N_1= \left \lfloor {\frac {N-1}{2}} \right \rfloor $
and
$N_2= \left \lfloor {\frac {N}{2}} \right \rfloor $
, where
$ \left \lfloor {\cdot } \right \rfloor $
is the usual floor function.
For
$x\in {\mathbb {Z}} $
, we define
$\widetilde {x}$
to be the unique integer such that
$\widetilde {x}\equiv x \mod N$
and
${-N_1\leq \widetilde {x}\leq N_2}$
.
To formulate the construction, we study the projective line over the finite ring
where
$(a,b)\sim (a',b')$
if
$(a,b)=u(a',b')$
for some unit
$u\in ({\mathbb Z}/N{\mathbb Z})^*$
. We write the class of
$(a,b)$
as
$(a:b)$
.
For
$x=(a:b)\in {\mathbb P}^1({\mathbb Z}/N{\mathbb Z})$
, define
The condition
$ma-b\in ({\mathbb Z}/N{\mathbb Z})^*$
says that
$(a:b)$
is distant from the affine point
$(1:m)$
in the usual terminology for projective lines over rings (see, e.g., [Reference Blunck and HavlicekBH01, Section 2]). This observation is only a useful way to recognize the condition and will not be used later.
Proposition 1.7 The function M in (1.6) is well defined and finite on
${\mathbb P}^1({\mathbb Z}/N{\mathbb Z})$
.
Let
$m=M(a:b)$
and put
Then,
$ (a:b)=(ac^{-1}:bc^{-1}) $
and
This gives a distinguished section of the natural projection from admissible pairs to
${\mathbb P}^1({\mathbb Z}/N{\mathbb Z})$
:
This is clearly independent of the choices of
$a, b$
in
$(a:b)$
, and we call
$\sigma (a:b)$
the M-normalized representative of
$(a:b)$
. If
$ \sigma (x)=(j,\ell ), $
then
Thus, every M-normalized representative with first coordinate j has the form
We introduce some notation:
where
$\pi _1:({\mathbb Z}/N{\mathbb Z})^2\to {\mathbb Z}/N{\mathbb Z}$
is projection to the first component. Therefore,
$\mathcal {S}(j)$
is the fiber of
$\pi _1:\mathcal {S}\to {\mathbb Z}/N{\mathbb Z}$
over j, and we aim to characterize it completely now.
Definition 1.12 For
$j\in {\mathbb Z}/N{\mathbb Z}$
, define
The set in (1.13) is nonempty, since
$Nj-1=-1$
in
${\mathbb Z}/N{\mathbb Z}$
. We regard
$W_j$
as the first break in the vertical string
The following proposition says that
$\mathcal {S}(j)$
is exactly the initial segment of this string before the break.
Proposition 1.14 For each
$j\in {\mathbb Z}/N{\mathbb Z}$
, we have
We define
Thus,
$M_j$
is the last height in the normalized vertical string over j, while
$W_j$
is the next height, where the string stops.
We now translate these projective-line representatives into matrices. Recall that
$\Gamma (1)$
is generated by
with
$S^2=-I$
.
The main result is the following theorem, where a fundamental list for
$\Gamma $
means, as in (1.3), a set of representatives for
$(\pm I)\Gamma \backslash \Gamma (1)$
.
Theorem 1.16 Let
$\Gamma (N)<\Gamma _1(N)<\Gamma _0(N)<\Gamma (1)$
.
-
(1) For $\Gamma _0(N)$
, a fundamental list of right coset representatives is
$\Theta _0$
given by (1.17) $$ \begin{align} \begin{cases} ST^i,& -N_1\leq i\leq N_2,\\ ST^jST^m,& -N_1\leq j\leq N_2,\ \gcd(j,N)>1,\ 0\leq m\leq M_j. \end{cases} \end{align} $$
-
(2) For $\Gamma _1(N)$
, a fundamental list of right coset representatives is
$\Theta _1$
given by (1.18) $$ \begin{align} \begin{cases} ST^i,& -N_1\leq i\leq N_2,\\ ST^jST^m,& -N_1\leq j\leq N_2,\ \gcd(j,N)>1,\ 0\leq m\leq M_j,\\ ST^kST^i,& -N_1\leq k\leq -2,\ \gcd(k,N)=1,\ -N_1\leq i\leq N_2,\\ ST^kST^{\widetilde{k^{-1}+j}}ST^m,& -N_1\leq k\leq -2,\ \gcd(k,N)=1,\\ &\qquad -N_1\leq j\leq N_2,\ \gcd(j,N)>1,\ 0\leq m\leq M_j, \end{cases} \end{align} $$where $k^{-1}$
denotes an inverse of k modulo N, and
$\widetilde {x}$
is defined in Notation 1.5.
-
(3) For $\Gamma (N)$
, a fundamental list of right coset representatives is $$\begin{align*}\{T^\ell\gamma\mid -N_1\leq\ell\leq N_2,\ \gamma\in\Theta_1\}. \end{align*}$$
Part (1), for
$\Gamma _0(N)$
, is the main point. The representatives for
$\Gamma _1(N)$
and
$\Gamma (N)$
are then obtained by the standard quotient maps between congruence subgroups.
In Section 2, we provide proofs for the results mentioned above. In Section 3, we give some examples and pictures.
2 Proofs of the results
Proof of Proposition 1.7
If
$(a,b)$
is replaced by
$u(a,b)$
with
$u\in ({\mathbb Z}/N{\mathbb Z})^*$
, then
$ma-b$
is replaced by
$u(ma-b)$
. Thus, the condition
$ma-b\in ({\mathbb Z}/N{\mathbb Z})^*$
is independent of the chosen representative of
$(a:b)$
, so M is well defined.
It remains to show that the set in (1.6) is nonempty. Let p be a prime divisor of N. If
$p\mid a$
, then
$p\nmid b$
by
$\gcd (a, b, N)=1$
, so
$ma-b\not \equiv 0\pmod p$
for every m. If
$p\nmid a$
, then the condition
$ma-b\not \equiv 0\pmod p$
excludes exactly one residue class of m modulo p. By the Chinese remainder theorem, choosing m to avoid these finitely many bad residue classes modulo the primes dividing N, we get
$ma-b$
prime to N. Hence, M is finite.
Proof of Proposition 1.14
For
$(j,\ell )\in \mathcal {S}(j)$
, we denote
$M(j:\ell )=m$
. Then, by (1.11),
$\ell = mj-1$
.
On the other hand, for
$m\in {\mathbb {Z}}_{\geq 0}$
, consider
$(j, mj-1)\in ({\mathbb Z}/N{\mathbb Z})^2$
. Suppose that
$\forall \, 0\leq t<m,\ tj - (mj-1)\notin ({\mathbb Z}/N{\mathbb Z})^*$
; then by
we know from (1.11) and (1.13) that
Therefore, we have the following equivalence:
We now record two elementary facts that convert the algebraic representative lists into fundamental domains.
Lemma 2.1 The set
described in (1.4) is a fundamental domain for
$\Gamma $
, provided that it is connected.
Proof Write
Since the tiles are ideal geodesic triangles,
so every nonempty open subset of E meets
$\mathcal D$
.
We check the conditions in Definition 1.1. For (1), openness and connectedness of E are immediate from the hypothesis. For (2), suppose
$x,y\in E$
and
$y=\gamma x$
for some
$\gamma \in \Gamma $
. Let
$U=E\cap \gamma ^{-1}E$
, a nonempty open set since it contains x. Since
$\mathcal D$
is dense and open in E, and
$\gamma ^{-1}\mathcal D$
is dense in
$\gamma ^{-1}E$
, we may choose
Then
$y'=\gamma x'\in \mathcal D$
. By the definition (2.2), write
$x'=\gamma _i x"$
and
$y'=\gamma _j y"$
, with
${x",y"\in D}$
. Since D is a fundamental domain for
$\Gamma (1)$
, we get
$x"=y"$
and
$\gamma \gamma _i=\pm \gamma _j$
. As the
$\gamma _i$
are representatives for
$(\pm I)\Gamma \backslash \Gamma (1)$
, this gives
$i=j$
,
$\gamma =\pm I$
, and hence
$x=y$
.
To prove (3), for any
$x\in \mathbb {H}$
, choose
$\theta \in \Gamma (1)$
and
$y\in \overline D$
with
$x=\theta y$
. Writing
${\theta =\pm \gamma \gamma _i}$
with
$\gamma \in \Gamma $
by (1.3), we get
$x=\gamma (\gamma _i y)$
and
$\gamma _i y\in \overline E$
. Thus, every
$\Gamma $
-orbit meets
$\overline E$
.
Definition 2.3 For a finite set
$\Theta \subset PSL_2({\mathbb {Z}})=SL_2({\mathbb {Z}})/(\pm I)$
, let
$G_\Theta $
be the graph with vertex set
$\Theta $
, where
$\theta $
and
$\theta '$
are adjacent if
Lemma 2.4 If
$G_\Theta $
is connected, then
is connected.
Proof If two vertices of
$G_\Theta $
are adjacent, the corresponding closed triangles share a full edge: after applying the inverse of one vertex, this is just the fact that
$\overline D$
shares its three edges with
$\overline {SD}$
,
$\overline {TD}$
, and
$\overline {T^{-1}D}$
. Choose a spanning tree for
$G_\Theta $
. Starting from one tile and adding the remaining tiles along this tree, each new tile is attached along a full edge, so the interior of the union remains connected. Hence, the full tile union has a connected interior.
Proof of Theorem 1.16
Part (1),
$\Gamma _0(N)$
. From [Reference CremonaCre97, Section 2.2], the right cosets in
$\Gamma _0(N)\backslash \Gamma (1)$
are in bijection with
${\mathbb P}^1({\mathbb Z}/N{\mathbb Z})$
. The bijection is induced by the bottom-row map
followed by passage to the projective class.
Let
Define
Thus,
$\mathrm {pr}$
keeps the usual affine representative on A and uses the M-normalized representative on H as in (1.10).
We compute
and
Therefore,
The first family represents the affine part A. The global fibers of
$\sigma $
are described by Proposition 1.14 for every residue class j. The auxiliary map
$\mathrm {pr}$
deliberately treats the affine part differently: it keeps these points in the usual affine coordinate, for the purpose of keeping the corresponding domain connected. The remaining non-affine points are exactly the fibers with nonunit first coordinate, and the second family with
$0\leq m\leq M_j$
represents them in their M-normalized form. Therefore, the bottom rows of the matrices in
$\Theta _0$
are exactly the pairs
$\mathrm {pr}(x)$
as x ranges through
${\mathbb P}^1({\mathbb Z}/N{\mathbb Z})$
.
It remains to prove connectedness. In
$G_{\Theta _0}$
, the vertex
$S=ST^0$
belongs to
$\Theta _0$
, and the vertices
$ST^i$
form the path
For each nonunit j, the vertex
$ST^jS$
is adjacent to
$ST^j$
, and the path
connects every vertex above j to
$ST^jS$
. Hence, every vertex of
$G_{\Theta _0}$
is connected to S. Lemmas 2.1 and 2.4 show that
$\Theta _0$
is a fundamental list for
$\Gamma _0(N)$
.
Part (2),
$\Gamma _1(N)$
. We use the exact sequence [Reference Diamond and ShurmanDS05, p. 14]
For a unit
$k\in {\mathbb Z}/N{\mathbb Z}$
, choose an inverse
$k^{-1}$
modulo N. Then,
and, as k runs through representatives for
$({\mathbb Z}/N{\mathbb Z})^*/(\pm I)$
, these matrices give representatives for
$(\pm I)\Gamma _1(N)\backslash \Gamma _0(N)$
. For the class of
$k=1$
, we use I instead of
${STSTS=-T^{-1}}$
. Thus,
is a set of representatives for
$(\pm I)\Gamma _1(N)\backslash \Gamma _0(N)$
.
Multiplying these representatives by the elements of
$\Theta _0$
gives the list
$\Theta _1$
in (1.18). The first two lines are
$I\cdot \Theta _0$
. The third and the fourth lines follow from
where we replace the exponents
$k^{-1}+i$
and
$k^{-1}+j$
by their symmetric representatives, as in Notation 1.5. We can do this, since the right cosets represented by
$ST^i$
and
$ST^jST^m$
in
$\Theta _0$
(1.17) depend only on the residue classes of i and j modulo N.
The graph
$G_{\Theta _1}$
contains
$G_{\Theta _0}$
. Since the allowed k’s lie in the range of the i’s,
$ST^kS$
is adjacent to
$ST^k$
, and
connects every vertex
$ST^kST^i$
to S. Similarly,
$\widetilde {k^{-1}+j}$
lies in the same range, and
connects every remaining vertex to S. Hence,
$G_{\Theta _1}$
is connected, and Lemmas 2.1 and 2.4 show that
$\Theta _1$
is a fundamental list for
$\Gamma _1(N)$
.
Part (3),
$\Gamma (N)$
. The matrices
$T^\ell $
, with
$-N_1\leq \ell \leq N_2$
, give representatives for
$\Gamma (N)\backslash \Gamma _1(N)$
. Therefore, the products
$T^\ell \gamma $
,
$\gamma \in \Theta _1$
, give representatives for
$(\pm I)\Gamma (N)\backslash \Gamma (1)$
. Since
$I=ST^0ST^0$
belongs to
$\Theta _1$
in
$PSL_2({\mathbb {Z}})$
, the vertices
$T^\ell $
occur in the list and form the path
For each fixed
$\ell $
, left multiplication by
$T^\ell $
gives a copy of the connected graph
$G_{\Theta _1}$
attached at
$T^\ell $
. Hence, the full graph is connected. Lemmas 2.1 and 2.4 finish the proof.
Remark 2.9 We consider a special case when
$N=p^rq^s$
has two prime factors p and q. For
$\gcd (a, N)>1$
,
$\gcd (b, N)>1,$
and
$\gcd (a, b, N)=1$
, we show that
$M(a:b)=1$
, that is,
$a-b$
is a unit mod N. This is true since under our assumption,
$\gcd (a, N)$
is one of the prime factors to some positive power and
$\gcd (b, N)$
is the other. Therefore,
$a-b$
is not divisible by either of the prime factors, so a unit mod N. Hence, M can be at most 1 in this case.
On the other hand, if
$N=p^r$
is a prime power, then
$\gcd (a, N)>1$
and
$\gcd (a, b, N)=1$
would require
$\gcd (b, N)=1$
, and then
$M(a:b)=0$
.
Meanwhile, the values
$M_j$
are not bounded by
$1$
in general; the first example with larger values is illustrated for
$N=30$
in Example 3.2 below.
3 Examples
We implemented the domains in Maple by drawing the ideal geodesic triangles. In the figures, Maple colors the edges red, blue, and green, with overlap priority red
$\,>$
blue
$\,>\,$
green.
Example 3.1 We show a labeled picture of our fundamental domain for
$\Gamma _0(6)$
(Figure 1). Since
and
$\phi (6)/2=1$
, this is also the domain for
$\Gamma _1(6)$
.
Our fundamental domain for
$\Gamma _0(6)$
labeled.
Figure 1 Long description
A mathematical plot on a Cartesian coordinate system where the x-axis ranges from negative 0.5 to 0.5 and the y-axis ranges from 0 to 1.2. The plot consists of various circular arcs and vertical lines forming a tessellation of the upper half-plane.
* At the top center, the largest region is bounded by a green arc and labeled with a large italic I.
* Below the I region, a central region is labeled with a large italic S.
* Flanking the central area are two large regions labeled S T on the left and S T super negative 1 on the right.
* Moving downward toward the x-axis, the regions become smaller and more numerous. On the left side, regions are labeled S T squared, S T squared S, S T cubed, S T cubed S, and S T cubed S T.
* On the right side, mirroring the left, regions are labeled S T super negative 2, S T super negative 2 S, and S T super negative 2 S T.
* The lines forming these boundaries are color-coded in red, blue, and green. All arcs converge toward specific points on the x-axis, most notably at x equals 0, x equals 0.5, and x equals negative 0.5.
The functions
$M_j$
and
$W_j$
are defined for every residue class
$j\in {\mathbb {Z}}/6{\mathbb {Z}}$
. The nonunit columns, which are the columns entering the second family of representatives in Theorem 1.16, are
The unit columns are part of the same globally defined functions, but the corresponding points are represented through the affine part of the list.
The connected picture makes several features visible at once. Internal base-side pairs
$(ST^jD,ST^jSD)$
form kites whenever both representatives occur; for instance, the
$j=0$
pair is
$(SD,D)$
because
$S^2=-I$
acts trivially on
$\mathbb {H}$
. Apart from the pairing of the left and right vertical sides, the bottom boundary arcs labeled
$1,\ldots ,8$
are paired by
as in [Reference NieNie26, Theorem 1.21]. The cusps are represented by
$\infty ,0,1/2,1/3$
, with widths
$1,6,3,2$
; there are no elliptic points of orders
$2$
or
$3$
, and the genus formula gives
$g(X_0(6))=0$
.
Example 3.2 The case
$N=30=2\cdot 3\cdot 5$
is the first one in which values
$M_j>1$
occur. Here,
The affine part of
$\mathbb {P}^1({\mathbb {Z}}/30{\mathbb {Z}})$
has
$30$
elements. The classes
$(j:-1)$
with nonunit j contribute
$22$
more, and the remaining
$20$
non-affine classes account for the higher values of
$M_j$
.
For nonunit
$j\in {\mathbb {Z}}/30{\mathbb {Z}}$
, the values are as follows. These are again only the nonunit columns of the globally defined functions
$M_j$
and
$W_j$
.
Thus, the number of representatives of the form
$ST^jST^m$
with j nonunit and
$0\leq m\leq M_j$
is
For
$ST^jST^m$
in (2.7), the associated cusp is
and it occurs
$M_j+1$
times. Grouping the above values by cusp equivalence gives the following widths; the contributing j’s are counted with multiplicity
$M_j+1$
.
These widths agree with the standard computations in [Reference Diamond and ShurmanDS05, Proposition 3.8.3] and [Reference SteinSte07, Algorithm 1.19]. This example is meant only to record the first occurrence of the higher values
$M_j>1$
; the general pattern is proved in the companion paper [Reference NieNie26].
Acknowledgments
The authors thank an anonymous referee for nice comments.
Open access
